Triangulation Operations

Adding or Clearing Ghost Triangles

An important component to be aware of when considering dynamic updates to triangulations are the ghost triangles. As we discussed in the vertex insertion/deletion example, ghost triangles are needed when we are making updates outside of the boundary of the current triangulation.

using DelaunayTriangulation
using CairoMakie

Let us take an example triangulation.

points = [(-1.0, -1.0), (1.0, -1.0), (0.0, 1.0)]
tri = triangulate(points)

Delaunay Triangulation.
   Number of vertices: 3
   Number of triangles: 1
   Number of edges: 3
   Has boundary nodes: false
   Has ghost triangles: true
   Curve-bounded: false
   Weighted: false
   Constrained: false

fig, ax, sc = triplot(tri, show_ghost_edges = true)
fig

The ghost triangles are represented by the convex regions bounded by the blue lines. By default, triangulate will keep these ghost triangles. If you want to remove them, you'd have to use delete_ghost_triangles! or use delete_ghosts = true inside triangulate.

If you do need to query whether your triangulation already has ghost triangles, use

DelaunayTriangulation.has_ghost_triangles(tri)

true

To clear the ghost triangles, use

delete_ghost_triangles!(tri)
DelaunayTriangulation.has_ghost_triangles(tri)

false

An important note for us to make is that ghost triangles are not just there as a concept, but they are actually physically stored. Adding them back with add_ghost_triangles!, we have:

add_ghost_triangles!(tri)
get_triangles(tri)

Set{Tuple{Int64, Int64, Int64}} with 4 elements:
  (3, 2, -1)
  (3, 1, 2)
  (1, 3, -1)
  (2, 1, -1)

See that there is not just the triangle (1, 2, 3), but also (3, 2, -1), (1, 3, -1), and (2, 1, -1) (where the ghost triangles are also oriented counter-clockwise). For example,

get_adjacent(tri, 3, 2)

-1

get_adjacent(tri, 3, -1)

get_adjacent(tri, -1, 2)

get_neighbours(tri, -1)

Set{Int64} with 3 elements:
  2
  3
  1

get_adjacent2vertex(tri, -1)

Set{Tuple{Int64, Int64}} with 3 elements:
  (3, 2)
  (1, 3)
  (2, 1)

If we delete them, they are no longer there.

delete_ghost_triangles!(tri)
get_triangles(tri)

Set{Tuple{Int64, Int64, Int64}} with 1 element:
  (3, 1, 2)

As a last note, we remark that the ghost vertices that define the vertex of these ghost triangles is still there regardless of whether the triangulation has ghost triangles. Thus, for example, the following still work

get_neighbours(tri, -1)

Set{Int64} with 3 elements:
  2
  3
  1

get_adjacent2vertex(tri, -1)

Set{Tuple{Int64, Int64}} with 3 elements:
  (3, 2)
  (1, 3)
  (2, 1)

get_adjacent(tri, 3, 2)

-1

You can remove them from the graph, using

DelaunayTriangulation.delete_ghost_vertices_from_graph!(tri)

Graph
    Number of edges: 6
    Number of vertices: 3

so that e.g. get_neighbours(tri, -1) is then an error. This will still not remove them from the adjacent and adjacent2vertex maps, but it does mean for example that

collect(each_solid_vertex(tri)) == collect(each_vertex(tri))

true

Just the code

An uncommented version of this example is given below. You can view the source code for this file here.

using DelaunayTriangulation
using CairoMakie

points = [(-1.0, -1.0), (1.0, -1.0), (0.0, 1.0)]
tri = triangulate(points)

fig, ax, sc = triplot(tri, show_ghost_edges = true)
fig

DelaunayTriangulation.has_ghost_triangles(tri)

delete_ghost_triangles!(tri)
DelaunayTriangulation.has_ghost_triangles(tri)

add_ghost_triangles!(tri)
get_triangles(tri)

get_adjacent(tri, 3, 2)

get_adjacent(tri, 3, -1)

get_adjacent(tri, -1, 2)

get_neighbours(tri, -1)

get_adjacent2vertex(tri, -1)

delete_ghost_triangles!(tri)
get_triangles(tri)

get_neighbours(tri, -1)

get_adjacent2vertex(tri, -1)

get_adjacent(tri, 3, 2)

DelaunayTriangulation.delete_ghost_vertices_from_graph!(tri)

collect(each_solid_vertex(tri)) == collect(each_vertex(tri))

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